Graphing Systems Of Equations To Solve Logarithmic Equations

Hey guys! Today, we're diving into the fascinating world of solving equations graphically, specifically focusing on logarithmic equations. It might sound a bit intimidating at first, but trust me, it's a super cool and effective method once you get the hang of it. We're going to break down a problem step-by-step, so you can see exactly how it works. Let's jump right in!

Understanding the Problem

So, the equation we're tackling today is:

$$\log _{0.5} x=\log _3 2+x$$

Now, at first glance, this might look a bit scary with those logarithms and different bases. But don't worry! Our mission is to figure out which system of equations we can graph to find the solution to this equation. Graphing is a fantastic way to visualize solutions, as it turns an algebraic problem into a visual one. We need to find two separate equations (y₁ and y₂) such that their intersection point on a graph will give us the solution to the original logarithmic equation. This involves cleverly manipulating the original equation into a form that's easier to graph.

Before we dive into the options, let's rewind a bit and make sure we're all on the same page about logarithms. A logarithm is essentially the inverse operation of exponentiation. When we see $\log_b a = c$, what we're really asking is, "To what power must we raise b to get a?" The answer is c. For example, $\log_2 8 = 3$ because 2 raised to the power of 3 is 8. Understanding this fundamental relationship is crucial for working with logarithmic equations. Another key concept is the change of base formula, which allows us to convert logarithms from one base to another. This is particularly useful when we want to graph logarithmic functions using a calculator or software that typically only supports common logarithms (base 10) or natural logarithms (base e). The change of base formula states that $\log_b a = \frac{\log_c a}{\log_c b}$, where c can be any base (usually 10 or e).

In our problem, we have logarithms with bases 0.5 and 3, which aren't the most common. This is where the change of base formula becomes our best friend. By converting both logarithms to a common base (like base 10), we can rewrite the equation in a form that's much easier to work with and graph. Remember, the goal is to isolate terms and create two separate equations, each representing a function that we can plot on a graph. The x-coordinate of the point where these two graphs intersect will be the solution to our original equation. So, with these logarithmic tools in our toolkit, we're well-equipped to dissect the given equation and figure out the best way to represent it graphically. Let's keep these concepts in mind as we analyze the options provided.

Analyzing the Options

We're given two options for systems of equations:

A. $y_1=\frac{\log 0.5}{x}, y_2=\frac{\log 3}{2+x}$ B. $y_1=\frac{\log x}{\log 0.5}, y_2=\frac{\log 2+x}{\log 3}$

Our task is to determine which of these, if any, correctly represents the original equation in a way that can be solved graphically. To do this, we'll need to manipulate the original equation and see which option matches the resulting system of equations. This involves applying the change of base formula, isolating terms, and thinking about how each side of the equation can be represented as a function.

Let's start by focusing on option A. The equations in option A look quite different from the logarithmic terms in our original equation. Specifically, the equation $y_1=\frac{\log 0.5}{x}$ seems to have inverted the logarithm of 0.5 and divided it by x. This doesn't align with any direct manipulation of the original equation using the change of base formula or basic algebraic operations. Similarly, $y_2=\frac{\log 3}{2+x}$ has a structure that doesn't immediately connect to the $\log_3 2 + x$ term in our original equation. There's a division by (2 + x), which isn't a natural outcome of applying logarithmic identities or transformations. Therefore, option A seems unlikely to be the correct representation.

Now, let's turn our attention to option B. The equations in option B, $y_1=\frac{\log x}{\log 0.5}$ and $y_2=\frac{\log 2+x}{\log 3}$, appear much more promising. Notice that $y_1$ looks like the result of applying the change of base formula to the left side of our original equation, $\log_{0.5} x$. The change of base formula allows us to rewrite $\log_{0.5} x$ as $\frac{\log x}{\log 0.5}$, which is exactly what we see in $y_1$. This is a strong indication that option B is on the right track. Now, let's examine $y_2$. It seems to be attempting to represent the right side of our original equation, $\log_3 2 + x$. However, there's a slight issue. The expression $\frac{\log 2+x}{\log 3}$ isn't a direct application of the change of base formula or a simple isolation of terms. It looks like there might be a misunderstanding in how the logarithm and the x term are being combined. To verify this further, we need to carefully break down the right side of the original equation and see if we can manipulate it to match the form of $y_2$. This detailed analysis will help us determine whether option B is indeed the correct system of equations or if there's a subtle error in its representation.

Solving the Problem

Let's take a closer look at our original equation:

$$\log _{0.5} x=\log _3 2+x$$

The key here is to use the change of base formula. Remember, the change of base formula states that $\log_b a = \frac{\log_c a}{\log_c b}$. We'll apply this to both logarithmic terms in our equation. It's usually easiest to change to a common base like 10, so we'll use that. Applying the change of base formula to the left side, $\log_{0.5} x$, we get:

$$\log_{0.5} x = \frac{\log x}{\log 0.5}$$

This perfectly matches the expression for y₁ in option B!

Now, let's tackle the right side of the equation, $\log_3 2 + x$. We'll apply the change of base formula to the logarithmic part, $\log_3 2$:

$$\log_3 2 = \frac{\log 2}{\log 3}$$

So, the entire right side of the equation becomes:

$$\frac{\log 2}{\log 3} + x$$

To combine these terms into a single fraction (which is what we see in the y₂ of option B), we need a common denominator. We can rewrite x as $\frac{x \log 3}{\log 3}$. Now we can add the fractions:

$$\frac{\log 2}{\log 3} + \frac{x \log 3}{\log 3} = \frac{\log 2 + x \log 3}{\log 3}$$

Wait a minute! This doesn't quite match the y₂ in option B, which is $\frac{\log 2+x}{\log 3}$. There's a crucial difference: in our derived expression, the x is multiplied by $\log 3$, while in option B, it's not. This means that option B is incorrect.

But don't lose hope! We've done the hard work of correctly rewriting the equation. We know that y₁ should be $\frac{\log x}{\log 0.5}$. And we know that y₂ should be $\frac{\log 2 + x \log 3}{\log 3}$. So, even though the given options weren't correct, we've successfully found the system of equations that could be graphed to solve the original equation. Sometimes, the process of elimination and careful manipulation leads us to the right answer, even if it's not explicitly presented in the options!

Conclusion

While neither of the provided options perfectly matched the system of equations needed to solve the given logarithmic equation graphically, we learned a ton in the process! We reinforced our understanding of logarithms, the change of base formula, and how to manipulate equations to make them graphable. Remember, the key to solving these types of problems is to break them down step-by-step, apply the relevant formulas and identities, and carefully compare your results with the given options. And even if the options aren't quite right, the process of working through the problem can lead you to the correct solution. Keep practicing, and you'll become a pro at solving logarithmic equations in no time! This journey highlights the importance of understanding the underlying mathematical principles rather than just memorizing formulas or guessing the right answer. By truly grasping the concepts, we can confidently tackle even the trickiest problems. Remember, math isn't just about finding the right answer; it's about the process of logical thinking and problem-solving.